Our pricing concept and approach have been discussed many times. For example, articles [1, 2, 3] were devoted to ConocoPhillips (NYSE: COP), which provides an excellent benchmark and general explanation of the quantitative definition of fundamental price. The measured COP price is accurately described by a linear combination of two consumer price indices, the core and headline ones. The headline CPI plays the role of that part of the overall energy price which is directly related to the share price. The agreement is so good that it is difficult to deny that energy (in one form or another) drives the evolution of ConocoPhillips.

where GEOI(t) is the share price in U.S. dollars at time t, CC is the core CPI, and E is the consumer price index of energy. The model standard error is only $2.87 for the period between July 2003 and March 2012. The model also allows both CPIs to lead the share price by 0 to 12 months. However, the best lags are both zero. The index of energy drives the price up, and the core CPI affects the price negatively. Figures 1 and 2 depict the observed and predicted monthly prices and the residual model error, respectively.

The overall agreement is good but the model residual has several spikes; the most recent has been observed since the second half of 2011. Currently, the error is +$6.5, which is an extremely high residual for GEOI. The same positive residual was observed a year ago. It had returned quickly to the predicted curve and then was followed by a negative residual of approximately the same amplitude. Therefore, the evolution of GEOI price is characterized by very high volatility likely associated with its size; smaller companies are subject to higher risks. At the same time, the COP model proves that the predicted price might play the role of “fundamental” price. Since GEOI is much smaller than COP, one can consider the current positive excursion as a short-term deviation. Then the GEOI price is highly overvalued.

Thus, we expect the GEOI price to fall to the level of $25 per share in the near future, i.e. we expect GEOI to return to its fundamental price.

Figure 1. The observed and predicted monthly closing prices for GEOI between July 2003 and March 2012.

In our previous articles on the Seeking Alpha [1, 2, 3] we demonstrated that the evolution of ConocoPhillips (NYSE: COP) share price can be accurately described by a model based on the share price decomposition into a weighted sum of individual consumer price indices (alternatively PPIs). Both defining CPIs (PPIs) may lag behind the price. This model has been working since 1982 with one structural break around 1998 [2]. This break was related to the change in the long term trend of the difference between the core and headline CPIs [4].

Figure 1 reproduces the observed and predicted COP prices in order to demonstrate the predictive power of the pricing concept. In Figure 2, we depict the model error between 1998 and 2012. One can see that the cumulative model error approached the zero line, i.e. the regression line coincides with the zero line. In other words, all deviations of the actual price from the predicted one are only short-term and the predicted curve might be considered as a “fundamental” one. For ConocoPhillips, we have estimated the following relationships to minimize the model error between 1998 and 2012:

COP(t) = 72.3 – 5.35(CC(t) - C(t)) (1)

where COP(t) is the share price in U.S. dollars at time t, CC(t) is the core CPI, and C(t) is the headline CPI. This relationship is slightly different from that in [2] where we did not minimized the model error.

Figure 2. The model error for COP. The regression line shows that the cumulative error approaches the zero line.

There is a good reason why we discuss the price model for COP over and over. It provides a good benchmark when we estimate quantitative models for different companies.With the predicted and measured COP prices in Figure 1, it is difficult to deny that energy (in one form or another) drives the evolution of ConocoPhillips. To many readers and/or investors this thought might be a trivial and obvious one. However, the thought that CPI components can drive share prices is not so trivial when we describe the prices of companies from different sectors and industries (e.g. BAC from Financial). The very same approach is treated as an inappropriate one.

In any case, there is no strong criticism of our models for energy related companies we continue presenting them with the case of Newfield Exploration Company (NYSE: NFX),an independent energy company, engaged in the exploration, development, and production of crude oil, natural gas, and natural gas liquids. As for many other companies, we tested two principal pairs of CPIs: C and CC; CC and the index of energy, E, as well as the pair the PPI and the producer price index of crude oil, OIL. The best fit (we used the LSQ technique) model is obtained with the pair CC and E:

where NFX(t) is the share price in U.S. dollars. We allowed both time lags (CPI leads the price) in (2) to vary between 0 and 12 months. However, the best lags are one and zero months, respectively. The index of energy drives the price up, and the core CPI affects the price negatively. Figures 3 and 4 depict the observed and predicted monthly prices and the residual model error, respectively.

The overall agreement is not good when compared to that for ConocoPhillips. Moreover, the model residual has been growing since the second half of 2011. Currently, the error is -$24.4, which is an extremely high residual. Now it’s time to use the COP model as a reference. In line with the interpretation of the COP model error, one might consider the predicted NFX price as a “fundamental” price. Since NFX is much smaller than COP its price is subject to higher fluctuations. Accordingly, the current excursion is just a short-term deviation. Then the NFX price is highly undervalued.

As an alternative explanation, one may suggest that the model has failed on NFX. We cannot exclude this explanation but then why the concept works for the biggest energy companies and has also been working relatively well before August 2011?Thus, we expect the NFX price to rise to the level of $60 per share in the near future, i.e. to return to its fundamental price.

Figure 3. The observed and predicted monthly closing prices for NFX between July 2003 and |March 2012.

Figure 4. The model residual error.

There is another interesting question why do the other two pairs of defining indices have larger residual errors? Obviously, all three oil related indices, i.e. the headline CPI, the index of energy and the producer price index of oil, are tightly linked but are also affected by different goods and services included in the CC and E. The evolution of these defining indices differs accordingly. On the other hand, there is some true set of goods and services which do define the evolution of the NFX price. Then the intercept of this true set and those comprising the CC, E, and OIL should mimic the behavior of the true defining set. In other words, all three studied indices are just proxy to the true one. As a result, their predictive power may vary with time.

3/30/12

Following our approach to share pricing of energy companies we present a model for Cameron International (NYSE: CAM). As always, the CAM model is based on our general concept that the evolution of any share price can be decomposed into a weighted sum (or difference) of two CPI/PPI components. Originally, this pricing concept was developed four years ago by predicting share prices for a few energy companies. We used the core CPI to represent an energy independent (but dynamic) reference to the headline CPI. The latter is used as a proxy to the economy-wide energy price, which is also related to oil price and the prosperity of energy companies. We assumed that the difference between these two CPIs might best express the energy pricing power relative to all other goods and services.

Cameron International Corporation provides flow equipment products, systems, and services worldwide. Here we demonstrate that the time history of a CAM share price, CAM(t), can be accurately approximated by a linear function of the difference between the core CPI, CC, and the headline CPI, C, in the United States. We test two more pricing models with the consumer price index of energy, E, and the difference between two PPI components: the overall PPI and the PPI of oil, OIL. Our goal is to find the model which describes the observed prices the best according to the LSQ criterion.

Three models below have been estimated together with their standard model errors for the same period between July 2003 and February 2012:

Model (3) provides the best explanation of the variability in the CAM share price since July 2003. The PPI of oil evolves in sync with the share, which grows proportionally to oil price. The PPI slope is negative and thus the growth in producer prices suppresses the growth in CAM price. Figures 1 through 3 clearly indicate that the closing price in February 2012 was estimated accurately and we do not expect any correction beyond the natural evolution according to (3).

Figure 1. The observed CAM price and that predicted from the core and headline CPI.

Figure 2. The observed CAM price and that predicted from the energy index, E, and the headline CPI.

Figure 3. The observed CAM price and that predicted from the PPI of oil, OIL, and the overall PPI. The high and low monthly prices represent the uncertainty of the observed price.

3/29/12

One of the commenters made a good remark that any quantitative model fails to predict for extended periods. In other words, the commenter doubts that our concept is valid and is able to provide an accurate description when the modeling period is extended back into the past. Fortunately, we have already addressed this issue in this article, which was also published by the Journal of Applied Research in Finances.

In order to explain our approach to stock pricing we have to start with an important observation on the difference between the core and the headline CPI. Four years ago we showed that there exist linear trends in consumer and producer price indices. Basically, it was found that the difference between the core CPI, CC, and the headline CPI, CPI, can be approximated by a linear time function:

dCPI(t) = CC(t) – CPI(t) = A1 + B1t

where dCPI(t) is the difference, A1 and B1 are empirical constants, and t is the elapsed time. Thus, the “distance” between the core CPI and the headline CPI is a linear function of time, with a positive or negative slope B1.

Figure 1 displays this difference from 1960 to 2012. There are three distinct periods of linear dependence on time: from 1960 to 1980, from 1981 to 1998, and from 2002 to 2008. The second period is characterized by a linear trend with slope B1=+0.65, and the third one has a larger negative slope of B1=-1.52. There are also two turning points or short time intervals - between 1980 and 1981, and from 1999 to 2002, where the trends undergo major changes. In 2008, we expected the difference to form a new linear trend, which would repeat the previously observed by duration and slope. In Figure 1, green solid line represents the expected trend between 2009 and 2015. Currently, there is some deviation from the expected trend. We believe that this deviation is a temporary one and the core CPI will soon to regain its pricing power over the prices of energy and food. For this, oil price has to fall to the level of $70 per barrel, which we observed in October 2011.

Figure 1. The difference between the core and headline CPI as a function of time. One can distinguish three periods of quasi-linear behavior with two distinct turning points. For second and third periods, linear regression lines are characterized by slopes B1=+0.65 and B1=-1.52, respectively. Green line represents the expected trend between 2009 and 2015, which we predicted as a mirror reflection of the previous trend.

This discussion is crucial for our stock pricing concept. It links the change in the difference to the change in pricing power. Apparently, when the difference turns to a trend with an opposite sign the pricing power of the relevant goods and services has to swop as well. As a result linear coefficients in our pricing model have to change their sighs as well.

The pricing model is a simple one. We assume the presence of a linear link between a stock price, say that of ConocoPhillips (COP), and the difference between the core and headline CPI,

COP(t) = A2 + B2dCPI(t)

where A2 and B2 are empirical constants, t is the elapsed time, and t2≥0 is the time delay between the stock and the CPI changes, i.e. the CPI may lag behind or lead the price.

We have already reported on the recent ConocoPhillips model and estimated all coefficients which best fit the observed price between July 2003 and February 2012:

COP(t) = -5.5dCPI(t) + 75

Figure 2 depicts the observed and predicted price since 1982. The above model does predict well after 1999, but there is no fit before 1999. Actually, the predicted and observed curves deviate spectacularly. At first glance, one might suggest that the dCPI provides no information about the evolution of the COP price.

The power of our pricing concept easily resolves this conflict. As we mentioned above, Figure 1 shows that the linear trend before 1999 was positive and after 2002 is a negative one. In terms of econometrics, there was a structural break in the dCPI behavior. The set of long-term economic links between goods and services, comprising the CPI and defining the linear trend in the dCPI between 1982 and 1999, underwent a three-year-long transition to a new set of links and constraints. In turn, this new set defines the trend observed from 2002 to 2008. It’s likely that the same trend is observed now. A reasonable assumption is that the sign of slope in the equation for COP(t) should also change to an opposite one. Since the positive slope between 1981 and 1999 is only between a half and one third of that between 2002 and 2008, one can expect that the slope observed before 1999 should also be divided by a factor of ~3.

After reversing the sign and calibrating relevant amplitude and level between 1982 and 1998 (we included the transition into the second segment) we have obtained a much better fit as depicted by green line in Figure 2:

COP(t) = 1.7dCPI(t) – 5; t between 1981 and 1998

Finally, a complete prediction of the COP price between 1982 and 2012 is obtained.

There is no special need to describe the price in the early 1980s using the CPI difference. All subcategories of the consumer price index, except the index for energy, are parallel before 1982. Therefore, the difference between any two indices, including the headline and core CPI, is constant, i.e. it contains no information on the changes in stock prices.

Figure 2. Historic (monthly closing) prices for COP (black line) and the scaled difference between the core CPI and the headline CPI (predicted price): green line from 1982 to 1998 and red line since 1998.

We expected that COP stocks will follow the new trend in the dCPI (green line) in Figure 1, as is did between 1985 and 2008, one will be able to predict the “trend price” at any given time before 2015. It did not happen yet and we are waiting for a turn to the new trend when oil price will go down. Meanwhile, any large deviation from the trend which will be compensated at a few year horizons might provide a good hint for short-term trading.

Overall, our pricing concept easily matched the challenge of the commenter. The difference between the core and headline CPI gives a good approximation to the evolution of COP price since 1982. There are short periods of rapid and deep fall in stock price which might be associated with the change in linear trends.

3/28/12

In Part 1 of this article, we discussed some similarities and differences in the evolution of the S&P 500 index before and after the 2001 and 2007 recessions. We have also shown that the current fall in the growth rate of working age population relative to that in 1990s and in the early 2000s cannot be responsible for the differences in the S&P 500 trajectories.In Part 2, we address the influence of labor force productivity on the stock market. There is always a question why should the growth in labor productivity accompany real economic growth as well as the stock market rallies?

Figure 1 reproduces the evolution of the S&P 500 index since 1982. An obvious feature of the curve is the presence of two peaks of the same amplitude in 2000 and 2007. We compared these peaks in Part 1 and tried (although failed) to relate them to the growth in working age population. Here we represent labor productivity in its differential form, i.e. as the rate of growth as reported by the Bureau of Labor Statistics.Therefore we also present the evolution of the S&P 500 index in the form of growth rate, which are usually called annual returns.From Figure 1, we calculate the relative S&P 500 change during a one year period. Since we use the monthly closing prices to represent the S&P 500 index, we also show the annual return at a monthly sampling rate.

Figure 2 shows the obtained return curve.The excellent agreement between peaks and troughs in Figure 1 is now destroyed because of small deviations in the slopes. Crudely, the overall behavior has many similar features as shown in Figure 3 where a copy (red line) of the original curve is shifted approximately six years and one quarter ahead in order to synchronize both peaks in the beginning of 2010.

Having clear similarities in the S&P returns during two previous recessions, we now can compare the evolution of the growth rate of labor productivity during the same periods. Figure 4 depicts the original curve (at quarterly rate averaged with MA(4)) and that shifted six years and one quarter ahead. There is no visible similarity between the curves and thus it is highly unlikely that the S&P 500 index has any traction with labor productivity. Hence, one should not consider labor productivity data when predicting the S&P 500 return.

Interestingly, the shape of the 1991/1992 recession fits much better the overall trajectory of the productivity growth during the 2007 recession. Moreover, the whole period between 2000 and 2010 is matched very well. It is a good question will the red curve continue to represent the future growth in labor productivity? It is worth noting once again that labor productivity does not drive the stock market.

Figure 1. The evolution of the S&P 500 index since 1982.

Figure 2. The evolution of the annual S&P 500 return at a monthly sampling rate.

Figure 3. Comparison of the 2001 and 2007 recessions. The original (black) curve was shifted ~6.25 years ahead (red curve) in order to synchronize both peaks in 2010. Crudely, the overall behavior during both recessions is similar.

Figure 4. The growth rate of labor productivity: the original curve (black) and that shifted six years and one quarter ahead (red line).

Figure 5. The growth rate of labor productivity: the original curve (black) and that shifted seventeen years ahead (red line).

3/27/12

It is difficult to deny that there exist general feelings of oil as a driver of economic growth. My own experience shows that any visible market analyst does not miss any opportunity to comment of changing oil price. An intriguing thing about these comments is that any big change in oil price is perceived as a danger for real economic growth, independent on the change sign. Skipping the question of oil influence on the entire economy, we would like to address the question of the driving forces behind the prices of energy companies.The pricing power of companies associated with Energy category of the S&P 500 list (we focus on this list in our articles) is of interest for many investors. Here we compare the price evolution of Chesapeake Energy (NYSE: CHK) and Chevron (NYSE: CVX).

In our previous post we presented a pricing model for Halliburton (NYSE: HAL). Also we have been routinely reporting on similar models for ConocoPhillips (COP) and ExxonMobil (XOM). All these models were based on our general concept that the evolution of a share price can be (quantitatively) decomposed in a weighted sum (or difference) of two CPI/PPI components. This concept was elaborated in a series of papers three years ago starting with the one predicting share prices for a COP and XOM since the early 1990s.

For Halliburton, we have exercised several models. The most basic model included the core and headline CPIs, where the core CPI, CC, was a proxy to an energy independent (dynamic) price reference and the headline CPI, C, played the role of energy price, which supposedly, moves the prices of energy companies. Therefore, we assumed that the difference between these two CPIs might express the energy pricing power relative to all other goods and services.As an alternative, we have tested two more pricing models for Halliburton with the core CPI the consumer price index of energy, E, and the producer price index of crude petroleum, OIL, together with the overall PPI. We have found that the best model for HAL includes CC and E with the standard model error of $3.55 between July 2003 and February 2012.

For this article, we have estimated three different models for share prices of two energy companies: Chesapeake and Chevron between July 2003 and February 2012. Figure 1 compares the evolution of actual monthly closing prices since 2003. The upper panel demonstrates that Chevron has a much better performance since 2009 and is currently at its historical peak ~$100; Chesapeake Energy has been fluctuating around $20. In the lower panel, we have displayed these prices as normalized to their respective peak values since July 2003. Chesapeake is now only at 34% of its peak in 2007. One might suggest that the difference in the overall behavior should be rooted in the driving forces behind these prices and thus can be caught in our models by the difference in defining CPIs.

All three studied models allow for different coefficients for defining CPIs, time lags and linear trend. The latter is an obvious component since we expect all share prices to rise with real economic growth.All three models are shown below with the standard errors estimated for the same period in Table 1:

CHK(t)= 3.68C(t) – 1.82CC(t-1) – 9.45(t-2000) – 185.01

CVX(t)= 2.07C(t) – 3.96CC(t-12) + 13.74(t-2000) + 208.52

CHK(t)= 172CC(t-6) + 0.39E(t) – 9.89(t-2000) - 235.34

CVX(t)= -2.65CC(t-12) + 0.25E(t) + 15.96(t-2000) + 277.77

CHK(t)= 1.11PPI(t) + 0.026OIL(t-12) – 6.75(t-2000) – 50.18

CVX(t)= 1.18PPI(t) – 0.044OIL(t-7) + 0.36(t-2000) – 138.25

Table 1. Standard model errors

Company

C and CC

CC and E

PPI and OIL

CHK

3.74

3.34

4.04

CVX

5.85

5.83

5.45

Figures 2 through 4 depict the observed and predicted prices for the three models. Table 1 implies (the lowermost standard model errors are highlighted) that the best CHK model is associated with the consumer price of energy and the core CPI. On the other hand, the price of CVX share is likely driven by the producer prices and, specifically, the price of oil. Therefore, the shares of energy companies not only demonstrate various time histories but are likely related to different goods and products. The investor could foresee the evolution of both prices if to project the defining CPIs and PPIs.

For CVX, the current price is at the expected level and any rise in oil price will likely reduce the price seven months later. The current price level of CHK is slightly undervalued. We will be reporting on both companies and expect an interesting comparison with oil price fluctuating in a wide range.

Figure 1. Relative performance of CHK and CVX. The upper panel shows actual prices and the lower one – the prices normalized to their respective peak values since 2003.

Figure 2.The observed CHK and CVX prices and those predicted from the core and headline CPI.

Figure 3.The observed CHK and CVX price and those predicted from the energy index, E,and the core CPI.

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Figure 4.The observed CHK and CVX prices and those predicted from the PPI of oil, OIL, and the overall PPI.

3/25/12

We have already presented our pricing model for Halliburton (NYSE: HAL). Also we have been routinely reporting on similar models for ConocoPhillips (COP) and ExxonMobil (XOM). All these models were based on our general concept that the evolution of any share price can be (quantitatively) represented as a weighted sum (or difference) of two CPI/PPI components. This pricing concept was developed four years ago by predicting share prices for a few energy companies. Essentially, we were trying to use the core CPI as an energy independent (dynamic) reference to the headline CPI which includes energy and, in turn, is related to oil price and thus the prices of energy companies. Therefore, we assumed that the difference between these two CPIs might be manifested in the energy pricing power relative to all other goods and services.At the later stages, the set of companies was extended to the S&P 500 list as a whole, and the model obtained new CPIs, time lags and a linear trend term.

ConocoPhillips and Exxon Mobil are the biggest energy companies and they have demonstrated almost the same sensitivity to the difference between the core, CC, and headline CPI, C, i.e. their pricing models were almost identical. Halliburton’s share price was also modeled and showed a different sensitivity to the change in the defining CPIs. We made a tentative conclusion that COP and XOM might have a larger return to the investor considering energy stocks.

Originally, we have demonstrated that the time history of a share price, p(t), (for example, HAL) could be accurately approximated by a linear function of the difference between the core CPI and the headline CPI in the United States. At the initial stage of our research, this difference was found to be the best to predict share prices of energy related companies. Mathematically, a share price, HAL(t), (we use a monthly closing price adjusted for dividends and splits) can be approximated by a linear function of the lagged difference between the core and headline CPI:

HAL(t) = 42 - 3.5dCPI(t+t1)(1)

where dCPI(t+t1)=CC(t+t1)–C(t+t1), t is the elapsed time, and t1=0 year is the time delay between the share and the CPI change.In the original model, the CPI difference had no time lag behind the share price, t1=0, and we covered the period between 1999 and 2009. The upper panel of Figure 1 shows the original model performance between July 2003 and February 2012, with the standard model error of $4.51.

Since 2009, the original model has been routinely validated by new data. Overall, it has demonstrated a good predictive power but we have also revealed some short-term deviations from observed prices.It was instructive to improve its performance and, using the experience with non-energy companies, to extend the set of defining CPIs. We have tested several pricing models for Halliburton with the same CPI and PPI components which were tested for ConocoPhillips and ExxonMobil. The extended set of defining indices includes: the core and headline CPI, the consumer price index of energy, E, and the producer price index of crude petroleum, OIL, together with the overall PPI. Thus, we tested models similar to (1) using two more differences for the period between 2001 and 2011. For this article, we have modeled the period between July 2003 and February 2012 with the same coefficients, which were obtained and reported in 2011:

HAL(t) = 30 - 0.30(CC - E); sterr =$4.85 (2)

HAL(t) = 25 - 0.13(OIL - PPI); sterr =$9.34 (3)

In other words, all coefficients in (1)-(3) were estimated by the least squares for the period between January 2001 and July 2011 and then used to describe the period through February 2012. The zero time lag was retained in the model similar to that in the ConocoPhillips and ExxonMobil models, where we found no time delay between the share price and defining differences. Unlike for COP, both standard model errors are larger than for the original model, i.e. the original model based on the headline and core CPI is the best among the three studied models. The upper panels in Figures 1 through 3 compare these three HAL models with zero time lags and no time trend. The modeling period is extended by seven months and the most important difference from the July 2011 is that the large excursion in the observed price has been well described by the dCPI. According to the best model, the current price is slightly undervalued.

At the same time, model (3) based on the producer price indices is the worst (sterr=$9.34). This may mean that Halliburton does not depend much on the producer prices. Interestingly, the change in oil price does accurately describe the period of the financial crisis. However, the model fails to predict slow changes in the share price. Model (3) has failed to predict the recent price peak but this deviation was only a transient one – the observed price is back to the predicted level.

Another possibility to improve the overall agreement between the observed and predicted prices is to allow for different coefficients for the defining CPIs, time lags and linear trend. The latter is an obvious component since we expect all share prices to rise with real economic growth. There models below have been estimated using these new features which have brought visible improvements as expressed by the standard model errors for the same period:

Model (5) provides the best explanation of the variability in the HAL share price since July 2003 using a time lead of only two months for the core CPI. The CPI of energy evolves in sync with the share. The CC slope is negative while the E slope is positive and thus we have the difference between the CPIs. The overall agreement between the observed and predicted prices is very good for the past nine years. The PPI model (6) is much better (sterr =$4.13 instead of $9.35) with time leads (OIL leads the share price by seven months) than model (3) without lags. Instructively, that the OIL slope is very small - the HAL share price does not depend on oil directly, but rather through the consumer price of energy as model (5) suggests. The dCPI model (4) is also better than (1) but has lost its position.

Finally, Figures 1 through 3 clearly indicate that the closing price in February 2012 was slightly undervalued and one can expect that the current deviation from the predicted price will disappear in the near future. The observed price may rise to $40-$42 per share in March/May 2012. The previous burst in price, observed between February and September 2011, has proved that the price quickly returns to the predicted level.

3/23/12

In our previous post we compared share price models for two financial companies from the S&P 500 list - Franklin Resources (NYSE: BEN) and Apartment Investment and Management Company (NYSE:AIV).There was a sound reason behind this direct comparison – both companies has similar models, i.e. are defined by the same consumer price indices. Here we extend our analysis by a triplet: Prudential Financial (NYSE: PRU), Loews corporation (NYSE: L) and MetLife (NYSE: MET). All three companies are driven by the change in the index of food and beverages (F) and the index of transportation services (TS). Several days ago we presented a model for Prudential on the Seeking Alfa. Therefore, this post is mainly devoted to L and MET.

Why do we rely on consumer price indices in our modeling? Many SA readers have reasonable doubts that some consumer price, which is not directly related to goods and services produced by a given company, may affect its price. We allow the economy to be a more complex system than described by a number of simple linear relations between share prices and goods. The connection between a firm and its products may be better expressed by goods and services which the company does not produce or provide. The demand/supply balance is fragile and may evolve along many nonlinear paths. It would be too simplistic to directly define a company price by its products.

So, the intuition behind our pricing model is more insightful – we link a given share to some goods and services (and thus their consumer price indices), which we have to find among various CPIs. In order to provide a dynamic reference we also introduce in the model some relative and independent level of prices (also expressed by CPIs). Hence, one needs two different CPIs to define a share price model. These CPIs we select from a predetermined set of 92 CPIs by minimizing the residual model error. All in all, we assume that any share price can be represented as a weighted sum of two consumer price indices (not seasonally adjusted in our model) which may be leading the share price by several months. Our model also includes a linear time trend and an intercept in order to remove mean and trend components from all involved time series.

The current Loews’ model is driven by the consumer price index of food and beverages, F, leading the price by seven months and the index transportation services, TS, which leads by five months:

The standard error between July 2003 and February 2012 is $3.25. It is worth noting that all coeffcinets and time leads are close and have the same signs, i.e. their influence of the corresponsing prices are similar.

Figure 1 displays the evolution of all three prices since 2003. Their shapes are mainly similar but the amplitudes are quite different. Figure 2 depicts the same curves but normalized to their respective peak values between 2003 and 2012. The overall similarity and the presence of a sharp fall in October-November 2008 are obvious. This might be the reason behind the same defining CPIs and close time delays of the share prices behind the CPIs. Apparently, any information about a probable fall in a share price obtained five months in advance could be of importance. That’s why we have been reporting on the prediction of our models for selected S&P 500 companies since 2009.

Figure 1. The evolution of PRU, L and MET share prices.

Figure 2. The evolution of PRU, L and MET share prices, all normalized to their peak values beteen 2003 and 2012.

We have already presented the newly estimated PRU model in our standard way. Figure 3 illustrates the evolution of both defining indices between 2002 and 2012. Figure 4 depicts the observed and predicted monthly closing prices for L and MET since 2003 and also provides an estimate of the models’ natural uncertainty as related to the high/low monthly prices. The real time prediction for L (red curve) leads the observed price by 5 month. As for PRU, all major turns in the prices were well foreseen by the models, including those in November 2007, March 2009, and April 2011. These dates are different from the pivot times for BEN and AIV. The respective residual errors are shown in Figure 5.

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